12.5: Bernoulli's Law

Objective

The student will:

Understand Bernoulli’s principle and be able to discuss its implications.

Vocabulary

Bernoulli’s Principle: A key principle connecting velocity and air. At those points in space where the velocity of a fluid is high, the pressure is low. At those points in space where the velocity of a fluid is low, the pressure is high.

Introduction

Flying has always been one of the great goals of human engineering. The key question is, what moves the force upward on an airplane wing? We can measure air speed and show that the shape of a wing makes the air go faster over the top of the wing than below the wing. How does this create lift, though?

A key principle connecting velocity and air was expressed by the Swiss mathematician Daniel Bernoulli (1700-1782). Bernoullo's Principle states:

At those points in space where the velocity of a fluid is high, the pressure is low. At those points in space where the velocity of a fluid is low, the pressure is high.

The mathematical equation that Bernoulli derived based on this principle is stated below.

Bernoulli’s Equation: P1+12ρv12+ρgh1=P2+12ρv22+ρgh2=constant

where P is the pressure of the fluid, ρ is the density of the fluid, v is the velocity of the fluid, and h is the height of the fluid. Though each term in the equation has units of pressure, Bernoulli derived the equation based on the conservation of energy. The equation is, in fact a statement of the conservation of energy.

A demonstration of this principle can be seen with a device called a venture tube, Figurebelow.

Notice in the Figureabove that where the fluid has a greater velocity, the vertical tube as a lower water level. This is the result of a lower internal pressure in the fluid at this point. If you have ever placed your finger partly over the opening of a garden hose, you’ve probably seen water velocity increase. The water sprays out! In the same way, in Figureabove, equal volumes of fluid must flow through the large cross sectional area (A1) and small cross sectional area (A2) of the tube. Therefore, the flow through the A2 must have a greater velocity than the flow A1.